Application Of Derivatives

Consider the following three statements for the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=\left|\log _e x\right|-|x-1|$ :

(I) $f$ is differentiable at all $x>0$.

(II) $f$ is increasing in $(0,1)$.

(III) $f$ is decreasing in $(1, \infty)$.

Then.